ixxss1 - Metamath Proof Explorer

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Theorem ixxss1 13396

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxss1.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})
ixxss1.3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))

**Assertion**
Ref	Expression
ixxss1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝐶,𝑥,𝑦,𝑧 𝑤,𝑂 𝑤,𝐵,𝑥,𝑦,𝑧 𝑤,𝑃 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝑥,𝑇,𝑦,𝑧 𝑤,𝑊 Allowed substitution hints: 𝑃(𝑥, 𝑦, 𝑧) 𝑅(𝑤) 𝑆(𝑤) 𝑇(𝑤) 𝑂(𝑥, 𝑦, 𝑧) 𝑊(𝑥, 𝑦, 𝑧)

**Proof of Theorem ixxss1**
Step	Hyp	Ref	Expression
1		ixxss1.2	. . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elixx3g 13391	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)))
3	2	simplbi 501	. . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
4	3	adantl 486	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
5	4	simp3d 1161	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ^*)
6		simplr 780	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵)
7	2	simprbi 502	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))
8	7	adantl 486	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))
9	8	simpld 499	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤)
10		simpll 778	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ^*)
11	4	simp1d 1159	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ^*)
12		ixxss1.3	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))
13	10, 11, 5, 12	syl3anc 1397	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))
14	6, 9, 13	mp2and 711	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤)
15	8	simprd 500	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶)
16	4	simp2d 1160	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ^*)
17		ixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 13387	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	10, 16, 18	syl2anc 595	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 14, 15, 19	mpbir3and 1360	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 417	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3942	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 {crab 3415 ⊆ wss 3904 class class class wbr 5108 (class class class)co 7412 ∈ cmpo 7414 ℝ^*cxr 11248

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-xr 11253

This theorem is used by: iooss1 13413 limsupgord 15530 pnfnei 23388 dvfsumrlimge0 26200 dvfsumrlim2 26202 tanord1 26713 rlimcnp 27141 rlimcnp2 27142 dchrisum0lem2a 27692 pntleml 27786 pnt 27789 liminfgord 46496

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